## CSC 221: Introduction to Programming Fall 2011

### HW 3: Volleyball Simulations

In class, we developed a series of Python functions for simulating the game of Pig and collecting statistics on game lengths (see control.pytxt for the code). For this assignment, you will develop similar functions for simulating volleyball games and collecting statistics on winning percentages.

Assume that the two teams in a volleyball game are each given a power ranking: an integer between 1 and 100. The higher the power ranking, the more likely that team is to win a particular point. For example, if team 1 and team 2 have identical power rankings, say 60 and 60, each is equally likely to win a given point. However, if team 1 has a power ranking of 80 and team 2 has a power ranking of 40, then team 1 is twice as likely to win a given point. The following Python function can be used to simulate a single point in a volleyball game.

1. Define a Python function named playGame that simulates a complete game of volleyball by calling playPoint repeatedly. The function has three inputs, the number of points required to win a game, the power ranking of team 1, and the power ranking of team2. It should print the score after each point and, at the end of the game, return the winning team number. For example:

Note that a team has to win a game by at least two points. You will need to think carefully how to address this in your function.

2. Once you have your playGame function working correctly, comment out the print statement by placing a '#' character at the front of the line. Then define a second function named playStats that performs repeated game simulations and keeps track of the winning percentages of the teams. The function should have four inputs, the number of games to be played, the number of points required to win a game, and the power rankings of the two teams. It should then repeatedly call the playGame function and keep track of the number of wins by each team. At the end, it should print a message of the form "Team 1 won X % of the games", where X is the winning percentage of team 1.

3. In our model, the power rankings provide a way of predicting which team will win a given point. A team with a 50% advantage in power rankings (e.g., 60 vs. 40) is 50% more likely to win a given point. However, this relative advantage may not hold the same over the course of an entire game. You are to conduct several experiments with different power rankings to determine how a relative advantage in points translates to a relative advantage in games. For each simulation, assume 10,000 games are to be played, with each requiring 15 points to win.

team1/team2 rankings expected % of points
won by team 1
actual % of games
won by team 1
50/50 50.0%
55/50 52.4%
60/50 54.5%
75/50 60.0%
80/40 66.7%
60/20 75.0%
80/20 80.0%

Describe the trend you see in the overall game wining percentages. Do they tend to match the point percentages fairly closely? If not, why do you think they differ? Is the effect consistent as the relative difference in point percentages increases?

4. Now, repeat your experiments, only with longer games. For each simulation, assume 10,000 games are to be played, with each requiring 25 points to win.

team1/team2 rankings expected % of points
won by team 1
actual % of games
won by team 1
50/50 50.0%
55/50 52.4%
60/50 54.5%
75/50 60.0%
80/40 66.7%
60/20 75.0%
80/20 80.0%

Describe the trend you see in the overall game wining percentages. Do they tend to match the game percentages from 15 point games fairly closely? If not, why do you think they differ? Is the effect consistent as the relative difference in point percentages increases?

Note: this work must be entirely your own, with no outside assistance other than the instructor.